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Definition

Given a vector space $V$, then its endomorphism Lie algebra $\mathfrak{end}(V)$ is the Lie algebra whose elements are the linear maps $\phi \colon V \to V$ and whose Lie bracket is their commutator

A Lie algebra homomorphisms $\mathfrak{g} \to \mathfrak{end}(V)$ is a Lie action or Lie algebra representation of $\mathfrak{g}$ on $V$.

Related concepts

• general linear Lie algebra